Summary: The Crust and the fiSkeleton: Combinatorial
Nina Amenta \Lambda Marshall Bern y David Eppstein z
December 2, 1997
We construct a graph on a planar point set, which captures its shape in
the following sense: if a smooth curve is sampled densely enough, the graph
on the samples is a polygonalization of the curve, with no extraneous edges.
The required sampling density varies with the Local Feature Size on the curve,
so that areas of less detail can be sampled less densely. We give two different
graphs that, in this sense, reconstruct smooth curves: a simple new construction
which we call the crust, and the fiskeleton, using a specific value of fi.
There are many situations in which a set of sample points lying on or near a surface
is used to reconstruct a polygonal approximation to the surface. In the plane, this
problem becomes a sort of unlabeled version of connectthedots: we are given a set of
points and asked to connect them into the most likely polygonal curve. We show that
under fairly generous and welldefined sampling conditions either of two proximity
based graphs defined on the set of points is guaranteed to reconstruct a smooth curve.
These two graphs are the crust, which we define below, and the fiskeleton, defined